This is a Phase II proposal to produce PDESYS, a software package for personal computers which provides an interactive graphical environment for defining, solving and viewing the solutions to systems of stationary and time-dependent elliptic partial differential equations (pdes) on arbitrary 1- and 2-dimensional spatial domains. The finite element method will be used to solve these equations on 2-dimensional domains while finite difference methods will be used for 1-dimensional domains. For 2-dimensional problems, the lower eigenvalues and eigenvectors of pde systems may be computed using the finite element basis and the Jacobi subspace iteration method. Partial differential equations (pdes) of the types to handled by the proposed software are ubiquitous in biomedicine, science and engineering. Existing packages for solving pde systems are large and difficult to use, and few of them are available for microcomputers. The proposal describes a system that is intended to be convenient, flexible and efficient. It is intended to be suitable for users who are less technically proficient in numerical analysis. What will distinguish this package from others is that it will emphasize convenience. PDESYS will use the MLAB modeling language as its interactive graphical shell.